A099889 XOR difference triangle of the odd numbers, read by rows.

1, 3, 2, 5, 6, 4, 7, 2, 4, 0, 9, 14, 12, 8, 8, 11, 2, 12, 0, 8, 0, 13, 6, 4, 8, 8, 0, 0, 15, 2, 4, 0, 8, 0, 0, 0, 17, 30, 28, 24, 24, 16, 16, 16, 16, 19, 2, 28, 0, 24, 0, 16, 0, 16, 0, 21, 6, 4, 24, 24, 0, 0, 16, 16, 0, 0, 23, 2, 4, 0, 24, 0, 0, 0, 16, 0, 0, 0, 25, 14, 12, 8, 8, 16, 16, 16, 16, 0

Offset: 0

Author

Paul D. Hanna, Oct 29 2004

Keywords

• nonn
• tabl

Comments

Main diagonal is A099890, the XOR BINOMIAL transform of the odd numbers. See A099884 for the definitions of the XOR BINOMIAL transform and the XOR difference triangle.

Examples

			Rows begin:
[1],
[3,2],
[5,6,4],
[7,2,4,0],
[9,14,12,8,8],
[11,2,12,0,8,0],
[13,6,4,8,8,0,0],
[15,2,4,0,8,0,0,0],
[17,30,28,24,24,16,16,16,16],...
		

Links

• Ivan Neretin, Table of n, a(n) for n = 0..8000

Crossrefs

Cf. A099884, A099890.

Programs

Mathematica

mx = 14; Flatten@Table[NestList[BitXor @@@ Transpose[{Most@#, Rest@#}] &, Range[1, 2 mx, 2], mx][[k, n - k]], {n, 2, mx}, {k, n - 1}] (* Ivan Neretin, Sep 01 2016 *)

PARI

T(n,k)=local(B);B=0;for(i=0,k,B=bitxor(B,binomial(k,i)%2*(2*(n-i)+1)));B

Formula

T(n, k) = SumXOR_{i=0..k} (C(k, i)mod 2)*(2*(n-i)+1), where SumXOR is the analog of summation under the binary XOR operation and C(k, i)mod 2 = A047999(k, i). T(2^n, 2^n) = 2^(n+1) for n>=0, with T(0, 0)=1.